Bases - What are they? (Part 2)

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Continued from the post about  ”What is a Base?”:

The same person wrote a follow up comment:

“you are not pretending i’m stupid!!!!! Okay is a base the number you can multiply by?????
“example: base two is 2,4,6,8,10,12,14,16,18 ?????????i don’t know what you mean!”

Yeah, keep trying to convince me that you’re stupid. From your grammar and your tone, you’re starting to make some headway. 

But I generally don’t believe a child can be stupid. Misguided, full of anxiety about themselves and the world, OK, but stupid is reserved for adults (where a lot of people make up for lost time). 

Maybe I wasn’t clear enough, so let me try again.  A base is a way to write a number using place value (columns). The amount of digits you decide to use in the columns determines the number of the base. If you use ten digits per column, the number will be in base 10. If you use three digits per column (the digits 0, 1 and 2), the number will be in base 3. You will understand this better as you read on.

You know what place value is. We use it in our daily number system. It’s the ones, tens, hundreds, thousands, etc. columns. Because the amount of digits we use in each column is 10 (those are the digits from zero to nine), we call our system the base 10 system. It’s also called the decimal system. The word decimal comes from the Greek word for ten - “deka.” 

Once you’ve used up the ten digits in a column, you must start filling up the next column. If you still don’t understand it, there is a good lesson about why we regroup and carry when we add at:

http://mathmojo.com/interestinglessons/regroupingandcarrying/regroupingandcarrying.html

That lesson explains what happens when we “go over” in a column, and why we use the next one. 

In the  base ten, as the numbers grow, each higher column is ten times the amount of the previous column. So in base 10, the columns go, ones, tens (because one times 10 is ten), hundreds (because ten times 10 is a hundred), thousands (because a hundred times 10 is a thousand), ten thousands (because a thousand times 10 is ten thousand), etc. 

In base 2, it would work the same way, except each time we would multiply the column by 2  (instead of 10) to get the next column. So in base 2, the columns go, ones, twos (because two times 1 is two), fours (because two times 2 is four), eights (because four times 2 is eight ), sixteens  (because eight times 2 is sixteen), etc.

We would need more columns to represent a number in a smaller base than in a larger one. Can you see why? 

It’s because If you had five columns in base two, the most you could represent with the fifth column would be 1 group of sixteen. 

If it was in base ten, by the time you got to the fifth column you could represent 9 groups of ten thousand! That is a lot more than sixteen!

In base 2, the columns go like this:   

By the way, the base 2 system is also called the “binary” system. “Bi” for “two,” as in bicycle, bisect, etc.

One thing you might be asking yourself, is, “What the heck is this stuff good for?” Fair enough question. 

Personally, I think the best thing this stuff is good for is to improve your mind, and learn how things work. Real, true, honest-to-goodness curiosity. Imagine that!

On a more everyday level, we use base 12 when we talk about dozens and grosses. When you pack things in dozens, you are giving the second place value a value of 12. And a gross it twelve twelves (or 122), which is 144. Base 12 is called the “duodecimal” system, from the Greek “dodeka.” (Can you se how that word looks like “dou deka?” or “two-ten?”  That’s where the English word “twelve” comes from.)

Base 2 happens to be the base which almost all computers function on. Computers don’t recognize anything but the digits 0 and 1. 0 means there is no flow of electricity, and 1 means there is flow. Like an on and off switch. 0 means off, and 1 means on. 

So why are there more numbers than that on a keyboard? Simple – every time you type a symbol in to a computer, the computer translates it to a base 2 number automatically, and that decides which switches are turned on and off, which makes certain things happen. 

(That is about the simplest explanation I can think of. Of course there is much, much more to it, but I hope it will do for now.) 

Since computers seem to be here to for the long-run, it pays to learn a little about programming, and learning base 2 is a good place to start. 

There are other reasons for learning other bases, and some of them have to do with how computers work, as well. For example computers use a modified base 16 (the hexadecimal system) to represent colors. As you know, there are millions and millions of colors in the world, and computers represent them with numbers. For example, the color bright red is #FF0000. (You’ll learn why we use letters with bases beyond 10 in a future post.) Using the hexadecimal system you can represent many millions using only six place values (columns). 

Cryptography (the science of codes) often makes use of bases other than 10. 

Many other  uses of other bases are too complex to explain, here, so I hope you will be patient and stick with math till you get to them. Some of them are about logic and decision-making, some are about better ways to do simple math (like using the abacus – base 5 has a lot to do with that), some are about geography and geometry (base 60, base 360), some are about history (base 20 and base 60 were commonly used), and some are about measurement. If you stay curious about the way things work, as you get more mature you will find lots of ways that bases are used. 

Believe it or not, I use this stuff for magic tricks. Some of the subtlest and best tricks that even fool other magicians are based on subtle math principles. Not the dumb kind like “take a number, multiply it by something, divide it by the number of coins in your pocket, add the number of teeth in your grandmother’s head,” etc.. but some great ones that are usually only seen and performed by elite magicians (and I don’t mean Chriss Whatshisface.)

Another good use for binary is in game theory. One good example is the game of Nimm. It’s sometimes used as a betting game, and if you are quick with using the binary system, you can win almost every time. I’ll teach it to you when we’re done with all the lessons about bases. That may take a few weeks, so make sure you learn all this stuff, and you’ll be ready to kick some butt at Nimm. 

I’ll also teach you how to count on your fingers in binary. You can use that skill to help you with huge additions in base 10. I know that sounds impossible, but if you learn it you can definitely become quicker than a calculator at large additions. 

Enough philosophizing for now! Let’s try converting one more number from base ten to base 2: 

Let’s try the number 7 (base 10):

A good rule of thumb, is to go through all the columns of the base you are converting to, until you reach a number that’s higher then the number you are converting. 

In other words, in this case we’, go through the powers of 2 until you reach a number higher than 7.

• The first column in base 2 is the ones column. 1 isn’t higher than seven, so keep going.

• The second column in base 2 is the twos column.  2 isn’t higher than seven, so keep going. 

• The third column in base 2 is the fours column.  4 isn’t higher than seven, so keep going. 

• The fourth column in base 2 is the eights column.  8 is higher than seven, so it is two much. You have to go back a column. 

This tells us that we’ll start writing the number 7 in base 2, starting with the fours column. We ask ourselves, “is there a 4 in 7?” There is, so we write a 1 in the fours column. Now we subtract the four from the 7 (because we’ve already used it in the fours column). That leaves us with 3.

Now look at the next column (the twos column). As yourself, “Is there a 2 in 3?” Yes, there is. Now we subtract the 2 from the 3 (because we’ve already used it in the twos column). That leaves us with 1.

And that goes in the ones column. 

So we have a 1 in the fours column, a 1 in the twos column, and a 1 in the ones column. That gives us the number 111 in base 2. 

That means that 7(base 10) is written as 111 (base 2). 

In the next post, we’ll learn how to read any base 2 number as a base 10 number. Let’s see if you can guess it before then. Think you can change 10101(base 2) into base 10? Leave the answer in a comment if you think you’ve got it. 

Also, for practice, turn 37 into base 2. Leave that in a comment as well, if you like.

Tags: convert base 10 to base 2 , what do we use bases for